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Arithmetic of 

W. Bro. Rev. F. de P. Cas tells, A.K.C. 

Originally published in 1914 


An Imprint Of Stephen A. Dafoe 

PO Box 5063 
Hinton, Alberta 
Canada - T7V 1X3 

Layout Copyright © 2005 by Stephen A. Dafoe 

All rights reserved. No part of this book may be reproduced or utilized in 
any form or by any means, electronic or mechanical, including 
photocopying, recording, or by any information storage and retrieval 
system, without permission in writing from the publisher. 

Library and Archives Canada Cataloguing in Publication 

Castells, F. de P. (Francis de Paula) 

Arithmetic of Freemasonry / F. de P. Castells. 

First ed. published: Leeds: Whitehead & Miller, 1914. 
ISBN 0-9731536-4-4 

1. Freemasonry-Symbolism. 2. Symbolism of numbers. I. Title. 
HS395.C33 2005 366M C2005-904003-3 

Printed In Canada 

Table of Contents 


Stephen A. Dafoe 

Pg. 04 


F. de P. Catells 

Pg. 05 


F. de P. Castells 

Pg. 06 

The Lecture: Arithmetic of Freemasonry 

F. de P. Castells 

Pg. 09 

Appendix One 

Metrological Symbolism of the Grave 

PS- *5 

Appendix Two 

Further Thoughts on the Evolution of Counting 

Pg. 26 

Appendix Three 

Some Thoughts on the Shem Ha-Meforash 

Pg. 28 

Appendix Four 

The Metrology of Solomon's Temple 

Pg. 29 


The Arithmetic of Freemasonry 


By W. Bro. Stephen Dafoe 

D espite the title of this booklet, the reader should not get the 
impression that they are about to read a dry account of 
“Arithmetic,” one of the Seven Liberal Arts and Sciences. 
While it does deal with the subject of Metrology, or the science of 
measurements, W. Brother Castells' book, “Arithmetic of 
Freemasonry” is a deeply esoteric work, dealing with a wide range of 
topic matter from how ancient civilizations developed systems of 
counting to the Kabalah and its connection to our Masonic Art. 

The author of this booklet, originally delivered the work you are 
about to read, as a lecture to the “Leeds Association of Installed 
Masters.” That lecture was delivered in 1914 , nearly a century ago, 
but the material herein contained has seldom been covered since 
that time. 

Sadly few Masonic books of this present era cover the esoteric 
topic matter of W. Bro. Castells' brief lecture on Masonic Metrology. 
Today Masonic authors, as scarce as we are, tend to focus on topic 
matters that require little thought and certainly no further back¬ 
ground study. 

For the Masonic student, who wishes to expand his knowledge of 
one of the Seven Liberal Arts and Sciences in an esoteric and spiri¬ 
tual direction, I am proud to offer this work, expanded with illustra¬ 
tions to illustrate the knowledge imparted by our departed, but not 
forgotten Brother, the Reverend F. de P. Castells. 

Stephen Dafoe 
June 24 th , 2005 


F. de P. Castells 


By F. de P. Castells 

T he leading ideas of this lecture first came to my mind when 
engaged in preparing a lecture on the Hebrew names for 
G.A.O.T.U. which I gave to the Bible Study Guide, organized 
by myself at Bexley Heath, Kent. Soon afterwards I developed those 
ideas and embodied them in a lecture on “The Hebrew Words of the 
Masonic Ritual,” which I delivered at the Lodge of Instruction 
carried on in connection with the North Kent Lodge, 2499. And last 
year, having joined the Lullingstone Lodge, 1837, Dartford, when 
invited to give a lecture there, I undertook to deal exclusively with 
the Numbers of the Ritual. Before coming to Leeds, the lecture was 
given to the Mid-Kent Masters' Lodge, 3173, Chatham, and that gave 
me a further opportunity of testing my opinions before giving them 
out as I do here. 


The Arithmetic of Freemasonry 


By F. de P. Castells 

A rithmetic is one of the “Seven Liberal Arts and Sciences” on 
which the Craft has been founded; it has been a potent factor 
in the development of the human mind, and may help us to 
understand some of the principles which underlie our Masonic 

Going back to primitive times, we find that there are two 
principle methods of counting: (l) By threes, and (2) by fives. The 
former leads us to the mysterious nine, and to the other sacred 
numbers of the ancient mysteries. The latter accounts for the ideal 
system of counting at the present time the Decimal system. 

The evolution of the Zero or Cipher: how it originated. Its 
powerful aid: it imports infinity into the calculations of the human 

The reasons for the adoption of the numbers 3 5 7 as elements of 
our symbolism, explained: 3 and 5 were adopted because Nature 
itself suggested them of mystic power. The number 7 is par 
excellence the Masonic number; its meaning is found by observing 
that it combines 3 and 4, thus 3 + 4 = 7. A review of the ceremonies 
and of the other matter contained in our Masonic Ritual will show 
that our ancient Brethren did not choose and of their symbolic 
numbers arbitrarily or at random. 

Some reflections on the hidden mystery of the summum 
numerorum, i.e., the composite 15. Its internal harmonies 
expounded and illustrated by the magic square and otherwise. 

The place assigned to this number 15, in the Masonic Ritual, an 
interpretation of the allegories - 

(a) The 30 F.C.'s of the traditional history. 

(b) The three figures 5 5 5 on the Tracing Board of the M.M., 
which are found to express an idea embodied in various ancient 
symbols, all of which agree with our story about the 30 F.C.'s 
The sacred seal of the G.G.O.T.U. arithmetically explained. 

The cabalistic significance of the Hebrew letter-numeral YOD 


F. de P. Castells 

Some considerations on the Divine Name JAH in its relation to 
the preceding facts. 

Didactic characters of the three elements 357 according to the 
actual statement of the Ritual. 

Some remarks on the Metrology of the Craft. 

Measures mentioned in the course of the ceremonies 
(a) The inch; (b) the foot; (c) the handbreadth; (d) the cubit; (e) the 
cable's length. 

The symbolism of the cable according to its construction. 

The dimensions of our last resting place; 3 feet wide, 6 feet long, 5 
feet deep or more, (see appendix one.) 


F. de P. Castells 

The Lecture 

By F. de P. Castells 

"hat we familiarly call Arithmetic has been, historically, 
one of the most potent factors in the development of the 
human mind; and as it is one of the Liberal Arts and 

Sciences on which Freemasonry has been founded, the stuffy of it 
should help us to understand some of the ideas underlying the Ritual 
of the Craft. The Architect, the Geometrician, and every Mason, 
whether operative or free and speculative, are absolutely dependent 
on the Science of Numbers. 

Long ago Pythagoras made his whole system of philosophy rest 
on the conception of numbers as regulating the relations of all 
things; and on this account he may be regarded as the philosopher of 
numbers. Philolaus, the earliest exponent of his teaching, put it in a 
nutshell, thus: “Number is the principle order, the principle on 
which the cosmos, or ordered world exists.” When 500 years ago, 
King Henry VI made inquiry concerning Freemasonry, he was told 
this: “It is the knowledge of nature and the power of its various 
operations, particularly the skill of reckoning, of weights and mea¬ 
sures, or constructing dwellings and buildings of all kinds, and the 
truer meaner of forming all things for the use of man.” And when he 
asked: “What do Masons conceal and hide?” his informer mentioned 
among other things: “The way of obtaining the faculty of ABRAC,” 
which is, it seems, just the sort of Arithmetic of which I am to speak to 
you in this lecture. Having regard to these facts we are justified, I 
think, in expecting that any numbers, quantities, and measurements 
mentioned in the Ritual must have a hidden significance. And, 
indeed, I am of the opinion that in the Ritual there is nothing arbi¬ 
trary, superfluous or meaningless. 

I am not aware that anyone has attempted to explain how the Art 
of Counting was evolved; but having travelled extensively and gained 
some knowledge of the ways of primitive races, I may venture to give 
you some of my own thoughts on the subject. At the dawn of civiliza¬ 
tion there seem to have been tribes counting “by threes” and tribes 
counting “by fives.” The trinal system must have been based on the 
intuitional conception which men had by nature of things and 


The Arithmetic of Freemasonry 

persons placed in three relative positions thus: — 

I - Thou - He. 

This - That - The other. 

One - Two - Three. 

But the quinary system was based on the five fingers of the hand, 
whence the name of digits still given to any integer under 10, a word 
which in Latin means “fingers.” As an illustration of the two systems 
let me take you on a journey Westwards, to the West Indies, and then 
afterwards to the mainland, Central America. The Caribbean people 
who dwell around the Bay of Honduras have only three indigenous 
words wherewith to express numerical values, viz.. Aba , Biama, 
Urua, meaning l 2 3. At present they count as we do, because they 
have learned the other numerals from the French missionaries who 
Christianized them; but when Columbus discovered the West Indies 
those three words, Aba, Biama, Urua were all the numerals they had. 
Then, if we quit the shores of the Bay of Honduras and travel inland 
into the interior of Central America we shall find that all the aborigi¬ 
nal tribes in that region follow the other system of counting, using 
their fingers and their toes in the operation, their name for 20 being 
Hu-winak, which means “one man,” because that number is the sum 
total of the fingers and toes on the human body. 

The crude system of the ancient Caribs may appear to us hope¬ 
lessly inadequate for any practical man of the present time; but of 
course the Caribs had ways of extending their calculations beyond 
three, saying for instance “twice two” for four; “tow and three” for 
five; “twice three” for six; “three times three” for nine; and if they 
had learned the use of markers for this high number, they would be 
able to count up to at least “three times nine,,” which yields 27, a sort 
of grand and Omnific number. It was through this method of count¬ 
ing among primitive men that when in course of time someone was 
able to devise a system of expressing numbers as figures, each of the 
nine digits came to be expressed by a separate character as is the case 
in our Arithmetic at the present day: 1 23456 7-8-9. 

The Greek imagination pictured Nine Goddesses, the Muses, 
who were said to be the daughters of Zeus and Mnemosune, that is, 
“God and the Memory,” each of them presiding over one of their 
“Nine Liberal Arts.” And in the Ancient Scottish Rite we find a Ninth 
Degree, in which the brethren are styled “Elected Knights of Nine.” 
Here, incidentally, I may call you attention to a curious property of 
“the Mysterious Nine,” which is that, whenever we add together the 
figures by which we express any of its multiples we always get either 


F. de P. Castells 

another nine or some lesser products resolvable to the same number. 
For example: 2 X 9 = 18; add 1 + 8 and you get 9; 4 X 9 = 36; add 3 + 6 
= 9; 11X9 = ; 99; 12X9 = 108; andsoon. 

The tribes that counted by threes and those that counted by fives 
developed their genius independently of each other; but we should 
not imagine that there ever was any race of men completely detached 
from the rest of the world or that owed nothing to the other races. 
There always was more or less inter-tribal commerce, and by that 
means men must have learned much from one another. For 
instance, in the barter of produce or merchandise, the Caribs, who 
always counted by threes, must have observed how easy it was to 
express the idea of five by showing “one hand” with extended fingers; 
ten by showing “both hands”; fifteen by presenting “two hands and 
one foot”; twenty by calling out “one man.” And having gone so far, 
they would see how easy it was to express forty by saying “two men,” 
and sixty by saying “three men.” May we not assume that it was in 
this way that the ancient Babylonians in Central Asia came to adopt 
their own sexagesimal system, using the number 60 (or “three times 
twenty”) in the same way as we use the 100? Then, too, by the 
exchange of produce, many must have found that, however dissimi¬ 
lar the two methods of counting may have appeared, they could 
easily be harmonized. Imagine for a moment, two groups of men 
engaged in arranging what they have brought for barter; one group 
count “two hands” and lay down ten articles eggs, melons, tubers, 
fowls, or as the case might be; but the other group counting by threes, 
lay down the wares they offer in piles of three, and then, for every 
complete tale of “three times three,” another article is “thrown in.” 
This extra something is placed either in front or by the side of the 
piles; and while to one party it helps to show at a glance the quantity 
of the goods involved in the transaction, to the other party it is an 
indispensable item for it turns the tale of nine into a full decade, 
completing the “two hands,” 9 + 1 = 10. It must have been in this way, 
through the intercourse of tribe and tribe, that the highly useful 
decimal marker came to be adopted; and then at length, somewhere, 
somehow, the use of this marker would also suggest the cipher of our 
arithmetical notation, the invention of which was needed for a great 
achievement, for, although the cipher when it stands by itself is but 
an empty symbol, when conjoined with any of the nine digits, may 
mean ten, hundreds, thousands of millions. The wonderful cipher or 
zero (however it may have originated) has led men to a great advance 
in the art of calculation. We may still call it nought, but it would be 
more appropriate to call it all, seeing that it imports infinity into our 
arithmetical operations. Our way of expressing the idea of TEN is 

_ 11 _ 

The Arithmetic of Freemasonry 

10, that is, a unit plus the decimal marker; and this unit signifies that 
on reaching ten we have begun a new series, v. gr. 9 + 1. to the deci¬ 
mal marker we have given the form of a circle, and this is quite 
fitting, too, for the circle is the emblem of eternity and infinity. I have 
illustrated the potentiality of the nine numerals when brought into 
combination with this circle in another of my diagrams, which shows 
the triple trine within the all-encircling orb. (See Appendix two.) 

What I have said thus far, explains the rise of two of our symbolic 
numbers, 3 and 5. But now, what about the number 7? This is a 
distinctively Masonic numeral, the mystic significance of which goes 
far deeper than that of the other two. It is a combination of 4 and 3, 
the square and the triangle, as we may see in the Ceremony of 
Raising by the steps which the candidate takes when advancing to 
the East, the first four describing a square in the c... of the L..., and 
the next three bringing him up to the Pedestal on which is the 
Volume of the Sacred Law. Beneath the ground so squared there is a 
yawning grave; but the candidate is a living being and his steps are 
emblematic at once of human life and of the four quarters of the 
globe. And as the last three steps indicate a triangle, which is symbol¬ 
ical of the G.A.O.T.U.. the two numbers 4 and 3, when conjoined or 
combined, suggest the thought of “God” brought into relations with 
his creatures and dominating human life.” I might enlarge on this, 
but it would lead me beyond the limits of our subject. 

Having explained the origin of the three symbolic numbers 357, 
we may now make a rapid review of the Ritual and see the function 
which each of them is made to discharge; and in this we ought to 
begin with the first, that is, number 3, which is recurring constantly 
in connection with (I.) our organization, (II.) the appointment of the 
Lodge, (III.) the traditional history, and (IV.) the actual mysteries or 
secret ritual acts. 

In our organization: 

Our Order stands on three grand principles, which are “Broth¬ 
erly Love, Relief and Truth.” 

There are three degrees, each degree with its respective signs, 
tokens and word. 

There are three principal, and three assistant, officers, the three 
principal ones occupying three different quarters of the globe, E., S. 

The distinguishing characteristics of a good freemason are three: 
Virtue, Honor and Mercy. 

There are three main lines of duty: to God, to our neighbor, and 

F. de P. Castells 

to ourselves; three sacred dictates: of truth, honor, and of virtue; 
three foremost excellences of character: secrecy, fidelity and obedi¬ 
ence; three lowest staves in the mystic ladder, which are said to be 
emblematic of the three graces, Faith, Hope and Charity; also there 
are :the three most celebrated orders of architecture, the Ionic, the 
Doric, and the Corinthian.” 

In the appointment of the Lodge: 

There are three sets of tools for each degree, that is, a different set 
of three tools for each one of the three classes of workmen. 

There are three Masonic reasons why the Lodge should be 
situated East and West. 

The interior of a Freemason's Lodge is composed of Ornaments 
Furniture and Jewels”; the Ornaments are three in number, viz. the 
Mosaic Pavement, the Blazing Star, and the Indented or Tessellated 
Border; the Furniture, too, comprises three articles, which are the 
Volume of the Sacred Law, the Square and the Compasses; while the 
Jewels are six, three moveable and three immovable. 

Similarly the Master Mason's Lodge is said to be distinguished 
by three Ornaments; the Porch, the Dormer and the Square 

The emblems of mortality are three: the coffin, the skull and the 

There are (we are told) three pillars supporting the Lodge: 
Wisdom, Strength and Beauty: there are also three greater lights and 
three lesser ones. 

In the traditional history: 

The first Lodge is said to have been consecrated with three grand 

The chapiters of the pillars, B. and J., are described as enriched 
with three varieties of ornamentation, viz., network, lily work and 
pomegranates; and these are said to symbolize Unity, Peace and 

The Entered Apprentices employed in King Solomon's Temple 
received a weekly allowance of Corn, Wine and Oil, which may be 
said to answer to what the present day laborer still regards as three 
necessities of life: Bread, Beer and Butter. 

At the time of the great Conspiracy there were but three men who 
possessed the secrets of the Master Mason. 

There were “three Grand Masters who bore sway at the building 


The Arithmetic of Freemasonry 

of the fir st Temple of J erusalem. ” 

The secrets of the Master could only be imparted with the con¬ 
sent and by the joint action of three. 

The trusty Fellowcrafts divided themselves into three 
Companies or Lodges. 

Our Master, H.A., was slain with three blows, struck by three 
different men, on three parts of the head, with three different tools, 
at three points of exit of King Solomon's Temple, situated on the S., 
N. andE. sides. 

In the mysteries: 

The Lodge opens and closes with three knocks and every candi¬ 
date gains admission by three knocks, given differently at each 

In every Degree the candidate who seeks the privileges corre¬ 
sponding to it by a three-fold plea: (I) by the help of God, being free 
and of good report; or (2) by the help of God and the assistance of the 
square and the benefit of the password; or (3) by the help of God, the 
united aid of the square and compasses, as well as the benefit of the 

On opening the Lodge the W.M. prays for three things: Order, 
Peace and Harmony. 

Every Initiate is made to strike the right sh... of the J. and S.W.'s 
with three knocks; he is made to advance to the pedestal by three 
steps; he has to answer three searching questions of the W.M.; he is 
told of the three dangers; the three reasons for soliciting his charity; 
the three aspects of his character, which are, “as a citizen of the 
world, and individual, and a Freemason.”; also of the threefold way 
of dividing the 24 hours of the day, viz., some for “prayer to Almighty 
God,” some for “labor and refreshment,” and some for “serving a 
friend or brother.” 

In the Second Degree the sign is of a threefold nature. 

In order to raise the candidate to the sublime degree of a Master 
Mason, three methods are tried; only the third can succeed. 

In closing the Fellowcraft Lodge, the dismissal formula, recited 
by the J.W. alone, is of a threefold nature. 

In the ordinary way of closing the Lodge, according to ancient 
custom, all the Brethren present unite in a threefold sign of F.F.F. 

The Master Mason, when raised, is asked to make his Solemn 
Obligation binding for as long as he shall live by sealing it with his 
lips three times on the Volume of the Sacred Law. 

We frequently have a threefold perambulation of the Lodge. 


F. de P. Castells 

The sign of g... and d... in Scotland, Ireland and the U.S.A. is 
given with a threefold movement of the hands accompanied with the 
threefold exclamation of “O.L. my G., etc.” 

This list does not exhaust all the references to the symbolic 3 
occurring in the Ritual, but it should suffice to show the important 
part that number plays in it. 

We may now pass on to consider the next number, 5. 

Three rule a Lodge, but five hold it, which means that there 
cannot be a lodge with less than five members. 

The Fellowcraft's of the traditional history form themselves into 
three Companies, that is three Lodges of five members. 

When the Entered Apprentice comes into a Fellowcraft Lodge, in 
order to be “passed,” he is taught to advance to the East by five steps. 

We acknowledge five Orders of Architecture. 

The Blazing Star is one of five points. 

There are five secret words, viz., those distinctive of the three 
degrees and the two passwords. 

There are five points of fellowship for us all, and these are said to 
correspond in number to the five signs of the Third Degree. 

The grave of our Grand Master was five or more feet deep. 

The chapiters on the pillars at the porchway or entrance of King 
Solomon's Temple were five cubits high. 

And now as regards the number 7: 

Seven is the number by which the Lodge is made perfect. 

In every properly constituted Lodge there are seven officers, that 
is, including the Tyler and Outer Guard. 

The first Tracing Board exhibits “seven stars, which have an 
allusion to as many regular made masons. ” 

In the ceremony of Raising, the method of advancing is by seven 

We are told that King Solomon's Temple was “seven years and 
upwards in building.” 

The height of B. and J. was 17 V2 cubits, which equals 7 + 7+3 V2, 
or the two pillars conjointly 35 cubits, which equals 5 times 7. 

The qualifications required from the would-be Master Mason 
are given thus: (1) Just, (2) Upright, (3) Free, (4) Men, (5) of Mature 
Age, (6) Sound Judgment, and (7) Strict Morals. 

The qualification of “Mature Age” is explained as meaning “the 
full age of 21 years,” which equals 3 times 7. 

Finally there are “seven Liberal Arts and Sciences” on which 
Freemasonry has been founded, viz., Grammar, rhetoric, Logic, 


The Arithmetic of Freemasonry 

Arithmetic, Geometry, Music and Astronomy.” 

But now, after this rapid survey of the Ritual, we shall find that 
these symbolic numbers, 357, important as they appear, are but 
fractions of what maybe called the summum numerorum, thus 3 + 5 
+ 7 = 15.1 hope to throw some new light on the mystic significance of 
this number; but first of all, I beg leave to call your attention to its 
internal harmonies. It has been said that “there is divinity in odd 
numbers”; and the number 15 is remarkable in many ways: 

It combines the first two of our symbolic numbers, for 3 x 5 = 15; 
and strangely enough the two numbers 3 and 5 joined together 
represent thirty-five, which equals 5x7, the other two symbolic 
numbers. Moreover, if we multiply the three numbers 3x5x7, the 
result is 105, their L.C.M., which also equals 7x 15, and is practically 
the same number 15, with the nought inserted between the figures. 

Chronologically, a period of fifteen days is that which is enclosed 
within three Sabbaths. Many of the ancient nations held their great 
Festivals on the 15 th day of a month; and as they usually followed the 
lunar year, the moon must at that time Have been full. 

In Greece, the Ceremonies of Initiation into the Eleusinian 
Mysteries (both in the Autumn and in the Spring) began on the 15 th 
day of the month, and lasted nine days. {Note these nine days, which 
equal “three times three. ”) 

In Music the number 15 represents two complete scales, that is, 
the notes comprised between three C's. 

As 5 stands for “one hand,” fifteen equals “three hands”; and 
there is somewhere a Chapter which has for its distinctive Badge a 
Triangle within which we see two hands stretched out from the sides 
and grasping each other, while a third hand descending from the 
apex holds up both of them. Although the interpretation of this 
symbol belongs to the Royal Arch, we may all read a beautiful 
thought into it, that of fraternal union on earth, maintained under 
the protection of the Most High. 

In the trinal system of primitive man, the numbers 6 and 9, 
which are the first trine doubled and trebled, were considered round 
numbers, and adding them together we get 6 + 9 = 15. 

Then, too, the number 15 combines the two leading terms of the 
decimal system, 10 + 5; which reminds us of the fact that the decimal 
system, now acknowledged to be the most perfect system of 
Arithmetic, is but the early method of counting by fives, that is to say, 
by the fingers of the hand. There really is nothing new under the sun. 

But suppose we take the first five digits as representing a pro¬ 
gressive arithmetical series, and that we add them up, 1+2+3+4+5, 


F. de P. Castells 




















again the total will be 15. and if we add up all the 
nine digits in the same way, we then bring the total 
to 45, which equals 3 times 15, and is half the 
number of the square 45 + 45 = 90. This I have 
illustrated by a diagram (right) which has a pile of 
bricks arranged in that same progressive order. 

Look for a moment at this magic 
square. In it, the nine digits are arranged in 
three rows, “three times three,: and 
however the figures may be added up, 
vertically, horizontally, or diagonally, the 
sum of every three is always the same, viz., 

15. The key to the various operations 
possible is to be found in the symbols from 
which we obtain the Masonic Alphabet. # 

X No other arrangements of the numbers 
will ever yield and other such harmony. 

Please observe that the central column consists of the symbolic 3 5 
7; the first at the bottom, 5 in the centre, 7 above; 5 stands between 1 
and 9, the unit which is the root of all number and 9 the highest digit; 
these two together complete the decade. 

My object in calling your attention to these facts is to show that 
the use of the number 15 in our Masonic symbolism is not arbitrary. 
It is the best our ancient could have chosen to convey the idea of 
fullness or completeness. Let us see, therefore, how it is that the 
number comes into the Ritual. 

I have said before that we open and close the Lodge with three 
knocks; but it would be equally correct to say that we open and close 
with 15 knocks, because the knocks prescribed by the Ritual are 
struck successively by the Worshipful Master, The Senior and Junior 
Wardens, and by both the Inner and Outer Guards, five officers in all 
and 5 times 3 make 15. 

Then we know how the candidate advances to the place of light 
by stages which are measured by a fixed number of steps: The 
Entered Apprentice takes 3; the Fellowcraft 5; the Master Mason 7. 
Add up all these and you have 15 as the total. This advancing order in 
the steps of each degree is foreshadowed by the Entered Apprentice 
at his initiation, for in his case the steps are of varying length, the 
second of a little longer than the first, and the third a little longer 
thanthesecond,asexpressedbythenumericalproportionof3 5 7. 

In the explanation of the Second Tracing Board we read that our 
ancient brethren “passed up the winding staircase, consisting of 3-5- 
7 or more steps”; and these correspond with the steps taken by our 

1 7 

The Arithmetic of Freemasonry 

candidates. And it should be observed that the phrase “or more 
steps” does not imply any uncertainty as to the height of the stair¬ 
case; it is merely a necessary reservation, the margin allowed for the 
requirement of the higher degrees. 

In the traditional history of the Third Degree we hear of a con¬ 
spiracy organized by “15 Fellowcrafts.” They were “of that superior 
class appointed to preside over the rest,” which means the order now 
denominated Master Masons. Out of the fifteen, twelve recanted but 
three proved to be “of a determined and atrocious character,” and 
were responsible for a great tragedy. As against these rebellious 
Fellowcrafts, there are mentioned another 15, who went to search for 
our Master, Hiram Abif; and we read that when these fifteen “trusty 
Fellowcrafts” were commissioned by King Solomon for their work, 
they divided themselves into three Companies, which means of 
course three Fellowcraft Lodges of five members each. One 
Company sought and found not; one Company made the discovery; 
and the third Company brought the evildoers to judgment. What 
then are we to think of the 30 Fellowcrafts? In my opinion they are 
emblematic of the struggle which is ever going on in the world 
between antagonistic forces with which every human being is more 
or less acquainted; the evil being ever envious of the good; the 
ambitious seeking to achieve success by deceit and rapacity, and in 
extreme cases by the most heinous crimes; the selfish and greedy 
thwarting and defeating the very ideals they profess to admire; all 
this in the hope of obtaining wealth, position or power. And 
Freemasonry steps in and counsels to substitute virtue, industry and 
perseverance for dishonesty, covetousness and violence, and to aim 
at the happiness of the fraternity. We may see the counterpart of this 
in the two intersected triangles shown on the Second Degree Tracing 
Board; one of the triangles is black and the other 
white, and the two are united by the letter G. 
inscribed in the common centre. I believe 
that the tragedy in the forefront of the 
Volume of the Sacred Law Cain 
slaying his brother Abel was intended 
to illustrate the same conception of 
life which is inculcated by 
Freemasonry. And the Chinese have 
from time immemorial represented 
the struggle which is ever going on; by 
the symbol they call Yin Yang, to be seen 
in one of the diagrams used to illustrate this 
lecture (right). 

_ 18 _ 

F. de P. Castells 

In the Tracing Board of the Master 
Mason there is another memorial of 
the death of our Master Hiram Abif; 
it takes the form of three fives 
arranged triangularly thus. 

They suggest an equilateral 
triangle, each 5 representing one 
side; but we know that histori¬ 
cally they correspond to the three 
methods tried for the purpose of 
bringing back to life the Ideal 
Man, our Master, Hiram Abif. The 
Junior Warden was the first to 
endeavor it, trying the g. of the E.A., 
which proved a slip; then the Senior 
Warden tried the G. of the F.C., 
which also proved ineffective; and 
then the W.M. applied the third 
method, which was successful. The 
three fives therefore are the three 
hands with fifteen fingers in all. There 
has been confusion over this, and some 
Rituals have been put forth that give 
the Fellowcrafts of the traditional 
history as twelve in number instead of 
fifteen. It seems to have arisen from 
certain distinctions being overlooked of 
forgotten. Let us remember that while 
the tradition mentions two Companies of 
15 Fellowcrafts, in each case, they are 
divided into 12 + 3; because one three of 
the rebellious Fellowcrafts are held 

responsible for the murder of our Master, and likewise his restora¬ 
tion to life is attributed to the action of three of the trusty 
Fellowcrafts. Three hands struck the fatal blows and three hands 
were exerted in raising the dead body; on this account both death 
and life may be expressed by three fives, or three hands. 

In closing the Fellowcraft Lodge, the Wardens report the discov¬ 
ery of “ a sacred symbol” said to be situated “in the centre of the 
building.” And this may be explained by what we read elsewhere: 
“When our ancient brethren were in the Middle Chamber of the 
Temple, their attention was peculiarly drawn to certain Hebrew 
characters, which are here depicted by a letter G., denoting God, the 


The Arithmetic of Freemasonry 

G.G.O.T.U., to whom we all must submit and whom we ought 
humbly to adore.” According to this, the cryptic G. is but our modern 
substitute for the Hebrew Characters depicted in the same Tracing 
Board just over the door at the top of the staircase, HT iT 1 . These 
characters are now commonly read Jehovah; but the name was 
always considered too sacred to be pronounced by human lips; and 
the Jews themselves in order not to write it have frequently 
expressed it with only the initial *• , the letter Yod. In speaking of this, 
however it is necessary to bear in mind that the characters of the 
Hebrew alphabet were numbers as well as letters, that is, they could 
be made to stand alternatively for a quantity or a sound. Here are the 
first ten of them, with their corresponding numerical value: - 

' am 1 rmat* 

10 987654321 

We see then, that the letter Yod^, the initial of the name Jehovah, 
when used numerically, represented the number 10, a fact which has 
afforded ground for much speculation. As previously shown, the 
Yod, standing for 10, may be regarded as emblematic of the perfect 
system of arithmetic. Some Jewish Rabbis (known as “The 
Cabalists”) pointed out that the Divine name in Hebrew consists of 
four letters (hence called the TETRAGRAMMATON or Four 
Lettered Name”), and by taking 1-2-3-4 as a series they reached a 
total of ten, which agreed with the face value of the initial. In this the 
Cabalists came in touch with Pythagoras and his “Holy Quaternion” 
in Greek Tetaktys), for Pythagoras held that Four the first square 
number was the potential decade, and it stood for the cosmic 
elements “fire, air, earth and water.” Sometimes the four Hebrew 
letters were arranged thus: - 


5 10 


1 n 1 

5 6 5 10 

6 5 to 


F. de P. Castells 

When combined the numerical value of the ten characters gave a 
total of 72, which the Jews regarded as an important symbolic 
number. (See Appendix three.) .) And sometimes in seeking to 
abbreviate the incommunicable Name of God they doubled or 
trebled the initial Yod, thus: - 

*» *1 *1 

~i *1 "l 

In each of these cases the result is a distinct triangle, but you may 
see that while the first is pointing down, the second is pointing up; 
and numerically, while the two Yods equal 20, the three equal 30. 

There is one more way of abbreviating the Ineffable Name, one 
which did not originate with the Cabalists, but is found in actual use 
in the Psalms, and may therefore be regarded as authorized by the 
Volume of the Sacred Law; I mean that which consists in writing the 
first two letters of the Name, H" 1 . (See Psalm lxviii. 4). Read phoneti¬ 
cally they spell Yah or Jah; but arithmetically they should represent 
the number 15, that is 10 + 5. The Jews used to express the numbers 
from ten to twenty by writing 10 + 1,10 + 2,10 + 3,10 + 4, etc.; but 
when they came to 15, they found that the two characters required 
were associated with the ineffable Name of God, and so they varied 
the order, writing, instead of 10 + 5,9 + 6. 

We have seen that in our Masonic symbolism, the number 15 has 
been made to embody the idea of fullness or totality; and now if we 
look again into the Ritual we may find a further illustration of this 
idea. For, in the explanation of the second Tracing Board, the 
Worshipful Master refers to the practical significance of the sym¬ 
bolic numbers 3 5 7, and reminds us that a Lodge is ruled by 3, held 
or organized by 5, and made perfect by 7. In this way those symbolic 
numbers suggest three leading ideas, viz., Government, 
Organization, and Perfect Development. And as the number 15 is 
reached by combining the three numbers, 3 + 5 + 7 = 15, we may 
regard this sum as comprehending all those ideas, and as being a 
fitting emblem of the Grand Ideal of Freemasonry. (See Appendix 
four.) In the Ancient Scottish Rite the members of the Tenth Degree 
are called “Illustrious Knights of Fifteen” as having attained to the 

_ 21 _ 

The Arithmetic of Freemasonry 

fullness of this ideal. 

I have undertaken to say something of the Metrology of the Craft, 
and I proceed to do so because it is a subject that stands closely allied 
with our arithmetic, but time compels me to be brief. The length 
measures mentioned in the Ritual are five: the two modern ones, the 
inch and the foot; the two ancient ones, the handbreadth and the 
cubit; and one which is very ancient although not a familiar one, viz., 
the cable's length. 

The word INCH comes from the Latin uncia, meaning “one 
twelfth part” of anything; and in this connection it indicates a twelfth 
part of a man's FOOT, which was early adopted as a length measure, 
being of course standardized everywhere. But in the Ritual, the 24- 
inch gauge is used as emblematic of time. When the Worshipful 
Master gives the Entered Apprentice his tools, he says that the 24- 
inch gauge represents the 24 hours of the day, and it may be used in 
two ways: to graduate his task and to divide each passing day so as to 
allow of his attending to everything: (i) his devotions, (2) his regular 
employment, and (3) the discharge of his social obligations. 

As the fingers of the hand have played an important part in the 
development of the art of counting, so, too, they supplied the basis 
for the art of mensuration all over the East; all the high metrical 
values consisting of so many palms or handbreadths, which like the 
foot, had to be standardized. In the explanation of the second 
Tracing Board we learn that the outer rim of the twin pillars of the 
porchway of king Solomon's Temple was “a HANDBREADTH in 
thickness.” This statement agrees with the 1 st Book of Kings (vi. 15). 
Elsewhere (Jeremiah lii. 21) we read that it was “four fingers” thick; 
and so I shall point out that in this there is no real contradiction, 
because the “handbreadth” of the Ritual is to be understood as being, 
not the breadth of the open or extended hand, but that of the back of 
a clenched hand, in which only four fingers are seen. 

Theoretically, the CUBIT was the length of the forearm from the 
elbow to the tip of the middle finger; but unfortunately as each 
nation adopted a different standard, we find a great deal of variety of 
cubits; indeed, some nations had two or three sorts of cubits accord¬ 
ing to the objects that had to be measured. This may explain the 
confusion and the controversy that there has been as to the actual 
length of the cubit employed in designing King Solomon's Temple. 
In Egypt, there were two different cubits, the short, which was one of 
six, and the royal, which was one of seven palms of handbreadths. In 
Babylonia there were three measurements known by that name: one 
of three palms, used in measuring fine decorative work and gold 
ornaments; one of four palms, which was used ordinarily for all 


F. de P. Castells 

masonry work; and one of five palms, which was used in measuring 
land-spaces. By the recovery of two scales of linear measurements 
engraved on the statues of Gudea, nearly 5,000 years old, we have 
found that these three cubits equaled 10.8, 14.4 and 18.0 inches 
respectively. Some archeologists argue that both King Solomon's 
Temple and in the Tabernacle, all three cubits of ancient Babylonia 
were used. On this calculation the thickness of the pillars would be 
equivalent to 3.6 inches of our present-day measure, although even 
now we cannot be quite certain. 

Finally about the CABLE'S LENGTH. When this expression is 
now used by sailors, they mean to indicate a distance of about one 
hundred fathoms, or the tenth part of a nautical mile. But as used in 
the Ritual, “at least a cable's length from the shore,” it means a 
distance equal to the length of a coil of cable, which has always been 
understood to be 120 fathoms, that is to say, 720 feet. The idea is 
that, being buried in the sands at the bottom of the sea at that dis¬ 
tance from low-water mark, as a minimum, the remains of a traitor 
might be considered to be definitely consigned to everlasting dis¬ 
grace, and therefore as having absolutely perished, “ no trace or 
remembrance: being any longer conceivable. This cable's length 
must not be confused with “the length of my cable tow” mentioned in 
another part of the Ritual, this latter being a figure of speech 
intended to convey the idea of one's ability to execute an acknowl¬ 
edged obligation, the power to fulfill one's own responsibilities in 
life. The introduction of the “cable” into the ceremonies may appear 
strange, but, apart from other considerations, if we only bear in mind 
the way the cable has been manufactured from time immemorial, we 
shall recognize that it is a very suitable emblem for the threefold cord 
which is not easily broken, of brotherly love, relief and truth. In a 
book compiled nearly one hundred years ago there is a description of 
it: “Every cable, of whatsoever thickness it be, is composed of three 
strands; every strand of three ropes; and every rope of three twists; 
the twist is made of more or less threads according as the cable is to 
be thicker or thinner. ” 

The Ritual anticipates that after this life every Master Mason 
shall be “decently interred”; but the penalty of our Obligation in the 
Third Degree involves the very opposite of that idea. I shall therefore 
close this lecture with a few words on the dimensions of the grave of 
our Master Hiram Abif, and their significance. I have spoken of 
fathoms, saying that a “cable's length” equals 120 fathoms; but the 
fathom is not named in the Ritual. It maybe defined as “the measure 
of a man's embrace,” fixed for all practical purposes at six feet; and 
therefore, although not expressly named, it is the length of the grave 

- 23 - 

The Arithmetic of Freemasonry 

which the trusty Fellowcrafts of our traditional history are said to 
have prepared. It was “from the centre 3 feet East and 3 feet West.” 
(See Appendix One.) This is an unusual form of expression, but we 
may account for it by observing that the navel is regarded as lying in 
the centre. Remember where the Master Mason rests his squared 
hand when he stands to order. The navel is our link with the past; and 
inasmuch as by being buried the body returns to the bosom of our 
common Mother, the old link is very properly viewed as the central 
point in the grave. The width of the grave “between North and South” 
was 3 feet. These dimensions are still the ordinary minimum dimen¬ 
sions of an adult's grave at the present time; but in the ritual they are 
given for a practical purpose, being emblematic. As the area was one 
of 18 square feet, 6 x 3 = 18 (nine to the East and nine to the West); 
and as the depth was a minimum of 5 feet, the total capacity of the 
grave was of 90 cubic feet, which is the number by which we define 
the Square “ an angle of 90 degrees.” The cubic measurement of the 
grave, therefore, is but another way of representing the Square. By 
being laid under the sod, back to the bosom of Mother Earth, we 
come into the closest possible fellowship, one with another, and the 
depth of 5 feet harmonizes with our five points of fellowship. In 
assigning such dimensions to the grave of our Master, Hiram Abif, 
therefore, the Ritual may be said to bring our symbolism into the 
reign of death; and we, by endorsing and adopting its provisions, are 
testifying to the fact that the bond which unites Master Masons in 
this life is not broken when we are let down into the tomb of trans¬ 
gression. Having ordered our life according to the Square, we take 
that Sign with us into the grave and we may rest assured that when 
we finally rise from the last resting place, the same old sign will 
procure our admission into the Grand Lodge above where the 
world's great Architect lives and reigns forever. 

24 - 

F. de P. Castells 

Appendix One 

Metrological Symbolism of the Grave 

F rom the ritual referred to by the author we find the following 
in reference to the grave of referred to in the Tracing Board: 

“Our Grand Master, Hiram Abif, was ordered to be interred 
as near the sanctum sanctorum as the Israelitish laws 
would permit, in a grave from the center three feet east, 
three feet west, three feet between north and south, and five 
feet or more perpendicular. ” 

The following illustration is provided to add further light to the 
author's comments on the size of the grave. 


5’ Deep = 5’ 

3’ between North and South = 3’ 

3’ East, 3’ West = 6 feet. 

6’ x 3’ x 5’ = 90 cubic feet. 


The Arithmetic of Freemasonry 

Appendix Two 

Further Thoughts on the Evolution of Counting 

I have spoken of what I observed among the Aborigines of Central 
America. In the South Sea Islands I discovered other facts which 
corroborate my views on the subject. In several of the Malay- 
Polynesian languages the words used for ten are modified forms of 
some terms alluding to an action in which both hands are shown. 
Thus, for instance, the word sangpoo, “ten” in Tagalog (Philippine 
Islands) is a contraction of sangpolan, which means “the whole”; and 
turo in the Pampango dialect means “the breadth of two hands.” Still 
it was long ages after men had learned to count by tens that the 
decimal system in its complete form as we have it now was evolved. 
What is really wonderful about this system is not its having a set of 
figures, from l to 9, but the fact that the figures are employed in such 
a way that by the aid of the zero, their value changes according to 
position. Take for example, 333 = 300,30,3. 

When primitive men wanted to make a record of their count, 
they would cut notches on a stick, and on reaching ten they would cut 
a cross like the Roman X; or if they had been marking the units by 
lines on any surface, after nine of these they would run another line 
across them to indicate the completeness of a decade. 

Our numerical figures are commonly said to be Arabic, because 
they were introduced into Europe by the Arabs; but those people did 
not invent them; indeed, in the time of Caliph Walid, A.D. 705-715, 
they had as yet no signs of numeration. The Arabs obtained them 
form India, where they had been in use for one thousand years before 
the Hegira, as is evidenced by the Nana Ghat inscriptions, which 
date from about 300 B.C. Presumably this method was suggested by 
the use of an abacus or counting-board, having an upper row of nine 
balls and a space beneath with one or more balls to indicate the tens, 
the hundreds and the thousands. The system in vogue in Ancient 
Egypt implies the existence of some such mechanical aid; for in the 
hieroglyphic inscriptions we find that the digits are expressed by 
plain strokes, nine being written thus, 111 111 111 (“three times three” 
once more), ten by a sort of inverted u, thus fl, which is repeated as 
required up to 90, and the hundreds by another character. 


F. de P. Castells 

As to the origin of the zero it has always puzzled investigators, 
but I am able to throw some light on it. Every authority I have 
consulted derives the word zero from the Arabic sifr, “empty,” 
through the Latin zephirus. The Arabic sifr has indeed so come to us 
in the form of Cipher; but I venture to say that the zero is a distinct 
word derived from the Hebrew oser, osarah, which means “ten.” The 
initial letter ain in the modern Hebrew script takes this form, AIN 
GOES HERE, but in ancient times, as we may see in the Moabite 
inscriptions of about 900 B.C., it took the form of a ring, whence the 
name ain, still given to the letter, which means, “an eye.” 


The Arithmetic of Freemasonry 

Appendix Three 

Some Thoughts on the Shem Ha-Meforash 

S ome of the Rabbis have taught that there are 72 angels or 
principalities who are the bearers of the “preeminent Name,” 
the Shem Ha-Meforash, which was arranged as shown below. 
These 72 angels are supposed to be alluded to in Exodus xiv. 
19,20,21, three verses which in Hebrew consist of seventy-two letters 

Exodusi4:ig 21 

19 And the angel of God, which went before the camp of Israel, 
removed and went behind them; and the pillar of the cloud went 
from before their face, and stood behind them: 

20 And it came between the camp of the Egyptians and the camp of 
Israel; and it was a cloud and darkness to them, but it gave light by 
night to these: so that the one came not near the other all the night. 

2 And Moses stretched out his hand over the sea; and the LORD 
caused the sea to go back by a strong east wind all that night, and 
made the sea dry land, and the waters were divided. 



T n 1 

n T n 

^ = 10 

n =05 
T =06 


F. de P. Castells 

Appendix Four 

The Metrology of Solomon’s Temple 

I n the measurements of King 
Solomon's Temple we 
constantly find multiples of 
the number 15. Here are some 
conspicuous instances: - 

The common interior height 
of the Holy Place up to the centre 
of the attic was 30 cubits, that is, 
twice 15. 

The breadth of the porchway 
or entrance was also 30 cubits. 

The circumference of the 
Molten Sea was 30 cubits. 

The joint interior length of the Holy Place and the Sanctum 
Sanctorum was 60 cubits, which is four times 15. 

The exterior width of the facade was 60 cubits. 

The total exterior length of the Sanctuary, including the Porch, 
was 90 cubits, six times 15. 

The total exterior height of the Porch, including the Middle 
Chamber and the Third Chamber, was 120 cubits, which equals eight 
times 15. (This is one-third of 360, the number of degrees into which 
we divide the circle, being equal to 4 x 90,12 x 30,24 x 15.) 

Between the Women's Court and the Court of Israel there were 
fifteen steps, and we are told that the worshippers used to sing the 
fifteen “Songs of Degrees” (Psalms cxx - cxxxiv.) while ascending 
those steps. 

Above all we should remember the 3-storied Chambers built for 
the Priests upon whom devolved the care of King Solomon's Temple. 
The tradition embodied in the Books of the Kings does not tell us how 
many there were; but Ezekiel gives us the number as 30; and we 
know that they were built on the three sides North, South and West. 
This number agrees with our tradition as to the 15 trusty and 15 
rebellious Fellowcrafts; also with the fact that when Samuel gave a 


The Arithmetic of Freemasonry 

Sacrificial Feast, he had 30 men to partake of it. Compare 1 st Samuel 
ix., 22; 1 st Kings vi., 5-8; 1 st Chronicles ix, 26; Ezekiel xli., 6. 

1 st Samuel 9:22 

22 Then Samuel brought Saul and his servant into the hall and seated 
them at the head of those who were invited about thirty in number. 

23 Samuel said to the cook, "Bring the piece of meat I gave you, the 
one I told you to lay aside." 

1 st Kings 6:5-8 

5 Against the walls of the main hall and inner sanctuary he built a 
structure around the building, in which there were side rooms. 

6 The lowest floor was five cubits wide, the middle floor six cubits 
and the third floor seven. He made offset ledges around the outside 
of the temple so that nothing would be inserted into the temple 

7 In building the temple, only blocks dressed at the quarry were 
used, and no hammer, chisel or any other iron tool was heard at the 
temple site while it was being built. 

8 The entrance to the lowest floor was on the south side of the 
temple; a stairway led up to the middle level and from there to the 
third. 9 So he built the temple and completed it, roofing it with 
beams and cedar planks. 10 And he built the side rooms all along the 
temple. The height of each was five cubits, and they were attached 
to the temple by beams of cedar. 

1 st Chronicles 9: 26 

26 But the four principal gatekeepers, who were Levites, were 
entrusted with the responsibility for the rooms and treasuries in the 
house of God. 

Ezekiel 41:6 

6 The side rooms were on three levels, one above another, thirty on 
each level. There were ledges all around the wall of the temple to 
serve as supports for the side rooms, so that the supports were not 
inserted into the wall of the temple. 


F. de P. Castells 




The Arithmetic of Freemasonry 

32 -